The name is absent

the lake to an oligotrophic state. Lakes that are deep and cold, or benefit from
rapid flushing or rates of sedimentation or that have been eutrophied for only
a short time will be easier to reverse (Carpenter et al., 1999). Shallow lakes,
however, will tend to be hysteretic due to characteristics such as warmer water
and higher ratio of water in contact with the bottom of the lake that confer
them higher rates of recycling.

The parameter values that characterize shallow lakes are identified as fol-
lows. Consider the following initial value problem given by the differential
equation (1) of section 2.1:

P(t) = L(t) - S^p(t) + r_p2P2+)m₂ ^{, p}(0) = P0^,            (2)

As in several economic analyses around the shallow lake, q = 2 is chosen as the
parameter for the steepness of the recycling response to the stock of phospho-
rous (Maler et al., 2003; Grüne et al., 2005; Wagener, 2003).

To make the problem scale invariant, the following substitutions are made¹ :

x/m,
ar,
br/m

and by changing the time scale to tr/m, one obtains the following equations for
the shallow lake dynamics:

2

a(t) = x(t) + bx(t)--5----

x²+1

This can be interpreted as the external loading of phosphorous as a func-
tion of the stock of phosphorous. It may seem strange from a biological point
of view to present loading as a function of the stock of phosphorous. However,
the objective of policy is to manage phosphorous content in the lake and, as
pointed out by Gruüne et al. (2005), “[t]he management can measure the stock
and can control the loading as a function of the stock”.

Assuming steady-state conditions where the stock of phosphorous is con-
stant, i.e. x(t) = 0, we have:

Furthermore, if the phosphorous loading is constant, i.e. da/dt = 0, we have:

a^(t) = b ^- (X⅛                   ⁽³⁾

¹Refer to Murray (1989) pp. 5 and 652 for a more detailed description of this technique.
Carpenter et al. (1999) also make use of it in Appendix A of their article, as do Maler et al.
(2003)

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